CHAPTER 1
Ab initio Calculation of Potential Energy Surfaces
BY R. F. W. BADER AND R. A. GANGI
1 The Concept of a Potential Energy Surface
This chapter is concerned with the calculation of potential energy surfaces by non-empirical methods, i.e. by obtaining solutions to Schrödinger's equation within the Born-Oppenheimer approximation. The concept of a potential energy surface is a consequence of the separation of the nuclear and electronic motions as proposed by Born and Oppenheimer. The nuclei may be considered to move under the influence of a potential determined by their mutual electrostatic repulsion and by the total energy of the electrons, an energy which is determined for every possible static configuration of the nuclei. The gain in conceptual simplicity afforded by the Born-Oppenheimer procedure is enormous and its use underlies many of our chemical concepts. For example, energies of activation or energy barriers in general, potential constants, the frequencies of vibrational and rotational motions, and bond lengths and bond angles as determined by an energy minimum, are all concepts defined in terms of the properties of a potential surface.
To obtain information concerning a system of n electrons and N nuclei in the absence of external fields, one must solve the time-independent Schrödinger equation
[FORMULA NOT REPRODUCIBLE IN ASCII] (1)
where the properties of the ith stationary state of the system of energy Ei are obtainable from the eigenfunctions Ψi(x,R) which in turn depend upon the space and spin co-ordinates of all the electrons and nuclei (whose collective co-ordinates are denoted by x and R, respectively) in the system. The total Hamiltonian operator is
[FORMULA NOT REPRODUCIBLE IN ASCII] (2)
where [??]e and [??N are the kinetic energy operators of the electrons and nuclei, respectively, and where the potential energy operator in atomic units is,
[FORMULA NOT REPRODUCIBLE IN ASCII] (3)
One can distinguish three levels of approximation in obtaining solutions to equation (1):2 (i) the Born-Oppenheimer approximation, often referred to as the clamped-nucleus approximation; (ii) the adiabatic approximation; and (iii) the non-adiabatic approximation. The solutions to equation (1) provided by the first of these are entirely adequate for most problems of chemical interest. For example, use of this approximation in the calculation of the low-lying vibrational and rotational levels of H+2 yields results which are correct within experimental accuracy.
In the Born-Oppenheimer approximation, one first transforms to a centre-of-mass, molecule-fixed co-ordinate system, thereby yielding a kinetic energy operator for the electrons referred to nuclei that are fixed with respect to their centre of mass and a nuclear kinetic energy operator which refers only to their internal motions, i.e. their vibrations and rotations and the various couplings between them.* Because of the large disparity in the masses of the electrons and the nuclei, the average kinetic energy of the former will be many times that of the latter, the ratio being proportional to mk/me ~ 2000 in the least favourable case when mk refers to the mass of a proton. Thus, in the limiting classical case, one obtains a picture of the electrons undergoing very rapid motion relative to the nuclei and adjusting almost instantaneously to changes in the nuclear positions. This suggests that the electronic motions are determined to a good approximation by just the static field of the nuclei, i.e. the electronic motion is determined by where the nuclei are but not by how fast they are moving. The motions of the nuclei in this separation of co-ordinates are governed by a potential whose negative gradient is simply the electrostatic force of repulsion between the nuclei and the attractive force exerted on the nuclei by the electronic charge distribution, a distribution whose form is determined by the electronic wavefunction evaluated for each configuration of the nuclei.
Mathematically, this separation of the kinetic motions of the electrons and nuclei amounts to approximating the total wavefunction Ψi (x, R) by a simple product of electronic and nuclear wavefunctions
[FORMULA NOT REPRODUCIBLE IN ASCII] (4)
The electronic function y>i(x; R) is obtained by solving the electronic Schrödinger equation for a fixed nuclear configuration,
[FORMULA NOT REPRODUCIBLE IN ASCII] (5)
where
[FORMULA NOT REPRODUCIBLE IN ASCII] (6)
The use of the semi-colon in the notation Ψi(x; R) is to denote the explicit dependence of Ψi on the electronic space-spin co-ordinates x and its implicit dependence, along with Ei on the nuclear co-ordinates R. Clearly, the solution of equation (5) for all spatial arrangements of the nuclei will generate an energy (hyper)-surface, the potential energy surface, which governs the motion of the nuclei as obtained by solving the nuclear eigenvalue equation
[FORMULA NOT REPRODUCIBLE IN ASCII] (7)
Equation (7) and its implied assumption of the separability of the electronic and nuclear motions is called the Born-Oppenheimer approximation.
The exact Schrödinger equation of motion, equation (1), may be equivalently stated in a manner which shows the neglected terms arising from the assumption of the product form for the wavefunction, equation (4). The exact eigenstate Ψi(x, R) is expanded in terms of the complete orthonormal set of functions Ψi(x; R) obtained from the solutions of the electronic equation, equation (5), in which case the nuclear wavefunctions χi (R) appear as the coefficients in the expansion. This procedure yields the following infinite set of coupled equations for the χi(R)
[FORMULA NOT REPRODUCIBLE IN ASCII] (8)
From this formal, but exact statement of Schrödinger's equation, which corresponds to the non-adiabatic approximation referred to above, one sees that the approximation made in obtaining equation (7) is to assume that
[FORMULA NOT REPRODUCIBLE IN ASCII] (9)
for all i and j. In another order of approximation, with the expansion of Ψi(x, R) truncated to the single product term as given in equation (4), one obtains in place of equation (8),
[FORMULA NOT REPRODUCIBLE IN ASCII] (10)
for the nuclear eigenfunction χiv(R). The term <Ψi|[??]NΨi> is called the adiabatic correction, a term which, as argued above, should contribute only 1/2000 or less than does Ei(R) to the total energy of the system.
Ko[??]os and Wolniewicz have obtained extremely accurate solutions to Schrödinger's electronic equation, equation (5), for H2 in its ground state over a large range of intemuclear distances. They further obtained solutions...
